Curve Transformation by n-Curving: Difference between revisions
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<math>\alpha(t)=\cos^{3}(2\pi t)+ i \sin^{3}(2\pi t)</math> |
<math>\alpha(t)=\cos^{3}(2\pi t)+ i \sin^{3}(2\pi t)</math> |
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The parametric equations of <math>\alpha\cdot u_{n}</math> are <math>x=\cos^{3}(2\pi t)+ \cos(2\pi nt)-1, y=\sin^{3}(2\pi t)+ \sin(2\pi nt)</math> |
The parametric equations of <math>\alpha\cdot u_{n}</math> are |
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<math>x=\cos^{3}(2\pi t)+ \cos(2\pi nt)-1, y=\sin^{3}(2\pi t)+ \sin(2\pi nt)</math> |
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See the figure.[[File:curve3.jpg]] |
See the figure.[[File:curve3.jpg]] |
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== n-Curving == |
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If <math>\gamma \in H</math>, then the automorphism <math>\alpha \to \gamma_{n}^{-1}\cdot \alpha\cdot\gamma_{n}</math> of the group ''G'', extended to the whole algebra ''C''[0, 1] is denoted by <math>\phi_{n}</math>. It is a linear operator that transforms every curve, for each value of ''n''. <math>\phi_{n}</math> is called an n-Curving. |
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== Example of n-curving == |
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==References== |
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[[Category:Algebras]] |
Revision as of 11:53, 29 October 2009
Introduction
There are many ways of transforming a mathematical curve. Here we introduce a method using the principles of Functional-Theoretic Algebra(FTA).
A curve γ in the FTA C[0, 1] of curves, is invertible, i.e.
exists if
.
If , then
.
The set G of invertible curves is a non-commutative group under multiplication. Also the set H of loops at 1 is an Abelian subgroup of G. If , then the mapping is an inner automorphism of the group G.
We use these concepts to define n-curves and n-curving.
n-Curves and Their Products
If x is a real number, then [x] denotes the greatest integer not greater than x and
If and n is a positive integer, then define a curve by
. is also a loop at 1 and we call it an n-curve. Note that every curve in H is a 1-curve.
Suppose Then, since , where
Example of a Product of n-curves
Let us take u, the unit circle centered at the origin and Products of n-curves often yield beautiful new curves. α, the astroid. Then,
The parametric equations of are See the figure.
n-Curving
If , then the automorphism of the group G, extended to the whole algebra C[0, 1] is denoted by . It is a linear operator that transforms every curve, for each value of n. is called an n-Curving.
Example of n-curving
References
- Sebastian Vattamattam, ``Transforming Curves by n-Curving, in ``Bulletin of Kerala Mathematics Association, Vol. 5, No. 1, December 2008